Recent Results on Edge-Colouring Graphs, With Applications to the Total-Chromatic Number

We give a brief account of some recent results on edge-colouring simple graphs and of some recent results on the total-chromatic number of simple graphs. We illustrate the kind of arguments which have been found to be successful by proving one of the simpler results on edge-colouring graphs, and by showing how to apply this to obtain one of the recent results on the total-chromatic number.